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Analytic birth—death processes: A Hilbert-space approach

Profile image of Markus KreerMarkus Kreer

1994, Stochastic Processes and their Applications

Abstract

Methods of Hilbert space theory together with the theory of analytic semigroups lead to an alternative approach for discussing an analytic birth and death process with the backward equations & = Ak_ ,g,_ ,-(/.L~ + hl)g, + pI+ ,gk+ I, k =O, 1,2,. ,., where A ~I = 0 = k,. Forrational growingforwardand backward transition rates A,=O(kY), pl=O(kY) (as k+m), with O<y< I, the existence and uniqueness of a solution (which is analytic for t > 0) can be proved under fairly general conditions; so can the discreteness of the spectrum. Even in the critical case of asymptotically symmetric transition rates A,-ficLI-kYone obtains for rational growing transition rates with 0 < y< 1 discreteness of the spectrum, generalizing a result of Chihara (1987) and disproving the traditional belief in a continuous spectrum. infinite tridiagonal matrices * discreteness of spectrum * analytic semigroups

Figures (2)
Proof. Since A: 4(A) > H is a closed symmetric operator with a compact resolvent R,(A) for a ¢>0 sufficiently large we conclude that the deficiency index is (0, 0) and thus A is self-adjoint (Kato, 1984, Chapter V, §3 Section 4, p. 271). As a self-adjoint operator with a compact resolvent A has a purely discrete spectrum consisting entirely of isolated eigen- values (Kato, 1984, Chapter III, §6 Section 8, p. 187). Furthermore A is negative semide- finite and therefore the unique generator of an analytic self-adjoint semigroup (Henry, 1981; notice that —A is sectorial in Henry’s notation). The formulas (9) and (10) are standard results in semigroup theory and functional analysis and so are the properties of A (Reed and Simon, 1980, Theorem XIII.64). Now for every given ¢>0 we can find NEN inde- pendent of t such that
Proof. Since A: 4(A) > H is a closed symmetric operator with a compact resolvent R,(A) for a ¢>0 sufficiently large we conclude that the deficiency index is (0, 0) and thus A is self-adjoint (Kato, 1984, Chapter V, §3 Section 4, p. 271). As a self-adjoint operator with a compact resolvent A has a purely discrete spectrum consisting entirely of isolated eigen- values (Kato, 1984, Chapter III, §6 Section 8, p. 187). Furthermore A is negative semide- finite and therefore the unique generator of an analytic self-adjoint semigroup (Henry, 1981; notice that —A is sectorial in Henry’s notation). The formulas (9) and (10) are standard results in semigroup theory and functional analysis and so are the properties of A (Reed and Simon, 1980, Theorem XIII.64). Now for every given ¢>0 we can find NEN inde- pendent of t such that

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